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A296295
Solution of the complementary equation a(n) = a(n-1) + a(n-2) + n*b(n), where a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5, and (a(n)) and (b(n)) are increasing complementary sequences.
2
1, 4, 15, 37, 80, 157, 291, 518, 897, 1523, 2550, 4227, 6969, 11417, 18638, 30340, 49298, 79995, 129689, 210121, 340290, 550936, 891798, 1443355, 2335825, 3779905, 6116510, 9897252, 16014658, 25912867, 41928545, 67842497, 109772194, 177615945, 287389465
OFFSET
0,2
COMMENTS
The increasing complementary sequences a() and b() are uniquely determined by the titular equation and initial values. a(n)/a(n-1) -> (1 + sqrt(5))/2 = golden ratio (A001622). See A296245 for a guide to related sequences.
LINKS
Clark Kimberling, Complementary equations, J. Int. Seq. 19 (2007), 1-13.
EXAMPLE
a(0) = 1, a(1) = 4, b(0) = 2, b(1) = 3, b(2) = 5
a(2) = a(0) + a(1) + 2*b(2) = 15
Complement: (b(n)) = (2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, ...)
MATHEMATICA
a[0] = 1; a[1] = 4; b[0] = 2; b[1] = 3; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + n*b[n];
j = 1; While[j < 10, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, k}]; (* A296295 *)
Table[b[n], {n, 0, 20}] (* complement *)
CROSSREFS
Sequence in context: A113289 A212974 A033813 * A296268 A209409 A241302
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Dec 14 2017
STATUS
approved